6.5 Conference Key Agreement with Single Photon Interference
99
E Z = p PE
b 1 =
N
i=2 b i |
M
j=1 k j = 1
(6.35)
E A 1 A i = p KG
b 1 = b i |
M
j=1 k j = 1
,
(6.36)
where p PE [] ( p KG []) is the probability that the event occurred in a PE (KG)
round. The asymptotic secret conference key rate achieved by the described CKA
protocol based on single-photon interference events reads:
r spCKA = Mp D
1 − h(E Z ) − max
2≤i≤N
h(E A 1 A i )
(6.37)
Notably, the above CKA scheme and the relative asymptotic key rate reduce to the
idealized TF-QKD protocol presented in Sect. 6.3.1 when N = M = 2.
6.5.1 Multipartite QKD with a W State
Here we would like to provide, as in the case of the idealized TF-QKD protocol, a
bit of intuition on why the above CKA protocol works.
When optimizing the conference key rate in (6.37) over q, we obtain values such
that 1 − q 1, that is the initial entangled states (6.32) are strongly unbalanced
towards the vacuum. Therefore, the rounds where only one detector clicked are
mainly caused by the sending and detection of just one photon. The photon could
be sent by any Alice i , implying that her qubit would be in state |1 A i while the
qubits of the other parties would be in state |0 A =i . Since the multiport BS creates a
coherent superposition of all these possibilities, by post-selecting the rounds where
e.g., detector D j clicked, the state of the parties’ qubits is approximately given by
the following W -class state
4 [31]:
|W j A 1 ... A N =
1
√
N
N
i=1
√
MU i j |v i ,
(6.38)
where the bitstring v i ∈ {0, 1}
N is composed of all zeroes except for the i-th bit
which has value one and where U i j is defined in (6.34). We remark that the qubits’
state in (6.38) is only an approximation in the limit 1 − q 1 where terms of second
or higher order in 1 − q have been neglected and in the ideal scenario of no losses
(
√
η = 1) and no other sources of noise.
Although it is proven [32] that no N -qubit state other than the GHZ state (4.1)
can lead to perfectly correlated outcomes in one measurement basis (for N ≥ 3), the
parties can still distil a secret conference key by properly measuring their qubits in
state (6.38).
4 The W state usually considered in the literature reads:
1
√
N
N
i=1 |v i .
99
E Z = p PE
b 1 =
N
i=2 b i |
M
j=1 k j = 1
(6.35)
E A 1 A i = p KG
b 1 = b i |
M
j=1 k j = 1
,
(6.36)
where p PE [] ( p KG []) is the probability that the event occurred in a PE (KG)
round. The asymptotic secret conference key rate achieved by the described CKA
protocol based on single-photon interference events reads:
r spCKA = Mp D
1 − h(E Z ) − max
2≤i≤N
h(E A 1 A i )
(6.37)
Notably, the above CKA scheme and the relative asymptotic key rate reduce to the
idealized TF-QKD protocol presented in Sect. 6.3.1 when N = M = 2.
6.5.1 Multipartite QKD with a W State
Here we would like to provide, as in the case of the idealized TF-QKD protocol, a
bit of intuition on why the above CKA protocol works.
When optimizing the conference key rate in (6.37) over q, we obtain values such
that 1 − q 1, that is the initial entangled states (6.32) are strongly unbalanced
towards the vacuum. Therefore, the rounds where only one detector clicked are
mainly caused by the sending and detection of just one photon. The photon could
be sent by any Alice i , implying that her qubit would be in state |1 A i while the
qubits of the other parties would be in state |0 A =i . Since the multiport BS creates a
coherent superposition of all these possibilities, by post-selecting the rounds where
e.g., detector D j clicked, the state of the parties’ qubits is approximately given by
the following W -class state
4 [31]:
|W j A 1 ... A N =
1
√
N
N
i=1
√
MU i j |v i ,
(6.38)
where the bitstring v i ∈ {0, 1}
N is composed of all zeroes except for the i-th bit
which has value one and where U i j is defined in (6.34). We remark that the qubits’
state in (6.38) is only an approximation in the limit 1 − q 1 where terms of second
or higher order in 1 − q have been neglected and in the ideal scenario of no losses
(
√
η = 1) and no other sources of noise.
Although it is proven [32] that no N -qubit state other than the GHZ state (4.1)
can lead to perfectly correlated outcomes in one measurement basis (for N ≥ 3), the
parties can still distil a secret conference key by properly measuring their qubits in
state (6.38).
4 The W state usually considered in the literature reads:
1
√
N
N
i=1 |v i .